Finite Projective Planes and Sharply 2-Transitive Subsets of Finite Groups

Lorimer, Peter

doi:10.1007/978-3-662-21571-5_43

Cited by 3 publications

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the 1970s, Lorimer started the classification of all finite permutation groups containing a sharply 2-transitive subset, see [6] for a summary of his work. In 1984, O'Nan continued Lorimer's research and mentions in [9] that "work on the groups Sp(2d, 2) of degrees 2 2d−1 ±2 d−1 is in progress" -these are the permutation groups our paper will mainly focus on.…”

Section: Sharply 2-transitive Sets In Finite Permutation Groupsmentioning

confidence: 99%

The non-existence of sharply 2-transitive sets of permutations in $$\mathrm {Sp}(2d,2)$$ Sp ( 2 d , 2 ) of degree $$2^{2d-1}\pm 2^{d-1}$$ 2 2 d - 1 ± 2 d - 1

Barth

2016

Beitr Algebra Geom

View full text Add to dashboard Cite

Abstract. We use Müller and Nagy's method of contradicting subsets to give a new proof for the non-existence of sharply 2-transitive subsets of the symplectic groups Sp(2d, 2) in their doubly-transitive actions of degrees 2 2d−1 ± 2 d−1 . The original proof by Grundhöfer and Müller was rather complicated and used some results from modular representation theory, whereas our new proof requires only simple counting arguments. Sharply 2-transitive sets in finite permutation groupsLet Ω be a finite set. A set S ⊆ Sym Ω of permutations is called sharply transitive if for each α, β ∈ Ω there is a unique g ∈ S with α g = β. We call S sharply 2-transitive on Ω if it is sharply transitive on the set of pairs (ω 1 , ω 2 ) ∈ Ω 2 with ω 1 = ω 2 .Sharply transitive and sharply 2-transitive sets of permutations correspond to Latin squares and projective planes, respectively, and are therefore relevant to the fields of combinatorics and geometry.In the 1970s, Lorimer started the classification of all finite permutation groups containing a sharply 2-transitive subset, see [6] for a summary of his work. In 1984, O'Nan continued Lorimer's research and mentions in [9] that "work on the groups Sp(2d, 2) of degrees 2 2d−1 ±2 d−1 is in progress" -these are the permutation groups our paper will mainly focus on. Later in 2009, Grundhöfer and Müller [5] extended O'Nan's character-theoretic methods and finally showed -among many non-existence results for other almost simple finite permutation groups -that there are no sharply 2-transitive sets in Sp(2d, 2) of degree 2 2d−1 ± 2 d−1 . Their complicated proof was based on results by Sastry and Sin [10] regarding characteristic 2 permutation representations for symplectic groups.In 2010, Müller and Nagy [8] introduced the so-called method of contradicting subsets and were -in addition to proving new results -also able to give simple combinatorial proofs for most of the previously known results by Lorimer, O'Nan, Grundhöfer and Müller.However, they didn't reprove the non-existence of sharply 2-transitive sets in the symplectic groups Sp(2d, 2) of degrees 2 2d−1 ± 2 d−1 . This is done in the present paper. More precisely, we construct contradicting subsets for the orthogonal groups O ± (2d, 2), d ≥ 4, in their natural actions on singular vectors -which happen to be the point stabilizers of the symplectic groups.

show abstract

Section: Sharply 2-transitive Sets In Finite Permutation Groupsmentioning

confidence: 99%

The non-existence of sharply 2-transitive sets of permutations in $$\mathrm {Sp}(2d,2)$$ Sp ( 2 d , 2 ) of degree $$2^{2d-1}\pm 2^{d-1}$$ 2 2 d - 1 ± 2 d - 1

Barth

2016

Beitr Algebra Geom

View full text Add to dashboard Cite

show abstract

Invertible Sharplyn-Transitive Sets

Polster

1998

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

We prove various results about sharply n-transitive sets of homeomorphisms of`n ice'' topological spaces like the real line and the circle. Our main results concern sharply 3-transitive sets G of homeomorphisms of the circle to itself. If G=G &1 , then G contains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharply n-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharply n-transitive sets of homeomorphisms have counterparts in the finite case.

show abstract

k-transitive permutation groups andk-planes

Barlotti

Strambach

1984

Math Z

View full text Add to dashboard Cite

Finite Projective Planes and Sharply 2-Transitive Subsets of Finite Groups

Cited by 3 publications

References 16 publications

The non-existence of sharply 2-transitive sets of permutations in $$\mathrm {Sp}(2d,2)$$ Sp ( 2 d , 2 ) of degree $$2^{2d-1}\pm 2^{d-1}$$ 2 2 d - 1 ± 2 d - 1

The non-existence of sharply 2-transitive sets of permutations in $$\mathrm {Sp}(2d,2)$$ Sp ( 2 d , 2 ) of degree $$2^{2d-1}\pm 2^{d-1}$$ 2 2 d - 1 ± 2 d - 1

Invertible Sharplyn-Transitive Sets

k-transitive permutation groups andk-planes

Contact Info

Product

Resources

About